Mathematical Modelling of Physical Systems 2021

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 11605

Special Issue Editor

Frankfurt Institute for Advanced Studies and Institute for Theoretical Physics, Goethe University, 60323 Frankfurt, Germany
Interests: mathematical modelling; classical and quantum gravity; quantum field theory; black holes; cosmology; theoretical particle physics; mathematical physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The majority of fundamental discoveries in physics have been obtained through a process of mathematical modelling—namely, the formulation and application of theoretical models and analytic methods for the description of a certain phenomenon. Mathematical modelling is used to address the foundations of potential disagreements between background knowledge and observed data. For example, this is the case of dark matter and dark energy, whose mysterious natures are challenging the standard model of particle physics and/or general relativity. Additionally, in the case of severe conflicts between two or more well-established formulations, a process of the model building must be invoked. This is the case for the combination of quantum mechanics with gravity and its related side effects—e.g., the hierarchy problem, the trans-planckian problem, the information loss paradox. Due to the focus on physical theories, mathematical modelling is based on mathematics that can be understood and used by theoretical physicists. Mathematical modelling can also consist of the application of such methods to hard sciences, including biology, finance, geology, climatology and engineering.

This Special Issue welcomes papers presenting major breakthroughs in modelling and drawing new scenarios. Please note that all submitted papers must be within the general scope of the Symmetry journal.

Prof. Dr. Piero Nicolini
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • physics beyond the standard model
  • alternative theories of gravity
  • semiclassical and quantum gravity
  • condensed matter physics and statistical physics
  • quantum information and artificial intelligence
  • dynamics of populations
  • pandemic
  • biological systems
  • econophysics
  • systematic risk
  • financial mathematics
  • dynamical systems and stability problems
  • methods of mathematical physics

Published Papers (8 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

Jump to: Review

25 pages, 3854 KiB  
Article
Omnidimensional Convex Polytopes
by Szymon Łukaszyk and Andrzej Tomski
Symmetry 2023, 15(3), 755; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15030755 - 19 Mar 2023
Cited by 2 | Viewed by 1478
Abstract
The study shows that the volumes and surfaces of n-balls, n-simplices, and n-orthoplices are holomorphic functions of n, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes [...] Read more.
The study shows that the volumes and surfaces of n-balls, n-simplices, and n-orthoplices are holomorphic functions of n, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects. In particular, for 0<n<1, the volumes of the omnidimensional polytopes are larger than those of circumscribing n-balls, and both their volumes and surfaces are smaller than those of inscribed n-balls. The surface of an n-simplex circumscribing a unit diameter n-ball is spirally convergent to zero with real n approaching negative infinity but first has a local maximum at n=3.5. The surface of an n-orthoplex circumscribing a unit diameter n-ball is spirally divergent with real n approaching negative infinity but first has a local minimum at n=1.5, where its real and imaginary parts are equal to each other; similarly, is its volume, where the similar local minimum occurs at n=3.5. Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about n-balls are proposed. Symmetries of products and quotients of the volumes in complex dimensions n and n and of the surfaces in complex dimensions n and 2n are shown to be independent of the metric factor and the gamma function. Specific symmetries also hold between the volumes and surfaces in dimensions n=1/2 and n=1/2. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
Show Figures

Figure 1

67 pages, 702 KiB  
Article
Modelling Cosmic Springs with Finsler and Generalised Finsler Geometries
by Matthew J. Lake
Symmetry 2022, 14(10), 2166; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14102166 - 16 Oct 2022
Cited by 1 | Viewed by 812
Abstract
We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a [...] Read more.
We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a (3+1)-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species, in a simple physically intuitive coordinate system, without the need for dimensional reduction or approximate effective actions. This, in turn, allows us to identify the precise conditions under which higher-dimensional strings and current-carrying defects are observationally indistinguishable, for macroscopic observers. Our approach naturally incorporates the description of so-called ‘cosmic springs’, whose dynamics are expressed in terms of an effective Finsler geometry, for circular loops, or generalised Finsler geometry, for non-circular configurations. This demonstrates the importance of these novel geometric structures and their utility in modelling complex physical phenomena in cosmology and astrophysics. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
41 pages, 479 KiB  
Article
Topological Quantization of Fractional Quantum Hall Conductivity
by J. Miller and M. A. Zubkov
Symmetry 2022, 14(10), 2095; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14102095 - 08 Oct 2022
Viewed by 856
Abstract
We derive a novel topological expression for the Hall conductivity. To that degree we consider the quantum Hall effect (QHE) in a system of interacting electrons. Our formalism is valid for systems in the presence of an external magnetic field, as well as [...] Read more.
We derive a novel topological expression for the Hall conductivity. To that degree we consider the quantum Hall effect (QHE) in a system of interacting electrons. Our formalism is valid for systems in the presence of an external magnetic field, as well as for systems with a nontrivial band topology. That is, the expressions for the conductivity derived are valid for both the ordinary QHE and for the intrinsic anomalous QHE. The expression for the conductivity applies to external fields that may vary in an arbitrary way, and takes into account disorder. Properties related to symmetry and topology are revealed in the fractional quantization of the Hall conductivity. It is assumed that the ground state of the system is degenerate. We represent the QHE conductivity as e2h×NK, where K is the degeneracy of the ground state, while N is the topological invariant composed of the Wigner-transformed multi-leg Green functions, which takes discrete values. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
15 pages, 373 KiB  
Article
A Constructive Method of Solving Local Boundary Value Problems for Nonlinear Systems with Perturbations and Control Delays
by Alexander N. Kvitko and Alexey S. Eremin
Symmetry 2022, 14(8), 1595; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081595 - 03 Aug 2022
Viewed by 886
Abstract
In this paper, a class of controllable nonlinear stationary systems of ordinary differential equations with an account of external perturbations is studied. The control satisfies given restrictions, and there is a fixed delay in it. An algorithm to construct a control transferring a [...] Read more.
In this paper, a class of controllable nonlinear stationary systems of ordinary differential equations with an account of external perturbations is studied. The control satisfies given restrictions, and there is a fixed delay in it. An algorithm to construct a control transferring a system from a certain initial state to an arbitrary neighborhood of the origin is proposed. The algorithm has both numerical and analytical stages and is easy to implement. A constructive sufficient Kalman-type condition of possibility of the transfer is derived. The algorithm efficiency is demonstrated by solving a robot manipulator controlling problem. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
Show Figures

Figure 1

10 pages, 1208 KiB  
Article
Impact of Brownian Motion on the Analytical Solutions of the Space-Fractional Stochastic Approximate Long Water Wave Equation
by Farah M. Al-Askar, Wael W. Mohammed and Mohammad Alshammari
Symmetry 2022, 14(4), 740; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14040740 - 04 Apr 2022
Cited by 15 | Viewed by 1557
Abstract
The space-fractional stochastic approximate long water wave equation (SFSALWWE) is considered in this work. The Riccati equation method is used to get analytical solutions of the SFSALWWE. This equation has never been examined with stochastic term and fractional space at the same time. [...] Read more.
The space-fractional stochastic approximate long water wave equation (SFSALWWE) is considered in this work. The Riccati equation method is used to get analytical solutions of the SFSALWWE. This equation has never been examined with stochastic term and fractional space at the same time. In general, the noise term that preserves the symmetry reduces the domain of instability. To check the impact of Brownian motion on these solutions, we use a MATLAB package to plot 3D and 2D graphs for some analytical fractional stochastic solutions. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
Show Figures

Figure 1

34 pages, 6638 KiB  
Article
Virtual Attractive-Repulsive Potentials Control Theory: A Review and an Extension to Riemannian Manifolds
by Luca Bigelli, Federico Polenta and Simone Fiori
Symmetry 2022, 14(2), 257; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020257 - 28 Jan 2022
Cited by 2 | Viewed by 2211
Abstract
The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, [...] Read more.
The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
Show Figures

Figure 1

13 pages, 286 KiB  
Article
Cylindrical Gravitational Wave: Source and Resonance
by Yu-Zhu Chen, Shi-Lin Li, Yu-Jie Chen and Wu-Sheng Dai
Symmetry 2021, 13(8), 1425; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081425 - 04 Aug 2021
Cited by 1 | Viewed by 1248
Abstract
Gravitational waves are regarded as linear waves in the weak field approximation, which ignores the spacetime singularity. In this paper, we analyze singularities in exact gravitational wave solutions. We provide an exact general solution of the gravitational wave with cylindrical symmetry. The general [...] Read more.
Gravitational waves are regarded as linear waves in the weak field approximation, which ignores the spacetime singularity. In this paper, we analyze singularities in exact gravitational wave solutions. We provide an exact general solution of the gravitational wave with cylindrical symmetry. The general solution includes some known cylindrical wave solutions as special cases. We show that there are two kinds of singularities in the cylindrical gravitational wave solution. The first kind of singularity corresponds to a singular source. The second kind of singularity corresponds to a resonance between different gravitational waves. When two gravitational waves coexist, the interference term in the source may vanish in the sense of time averaging. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

Review

Jump to: Research

28 pages, 593 KiB  
Review
A Critical Review of Works Pertinent to the Einstein-Bohr Debate and Bell’s Theorem
by Karl Hess
Symmetry 2022, 14(1), 163; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010163 - 14 Jan 2022
Cited by 11 | Viewed by 1527
Abstract
This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems [...] Read more.
This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems of Bell and Clauser, Horne, Shimony and Holt (CHSH) deal with mathematical abstractions that have only a tenuous relation to quantum theory and the actual EPRB experiments. It is also shown that, therefore, Bell-CHSH cannot be used to assess the nature of quantum entanglement, nor can physical features of entanglement be used to prove Bell-CHSH. Their proofs are, among other factors, based on a statistical sampling argument that is invalid for general physical entities and processes and only applicable for finite “populations”; not for elements of physical reality that are linked, for example, to a time-like continuum. Bell-CHSH have, furthermore, neglected the subtleties of the theorem of Vorob’ev that includes their theorems as special cases. Vorob’ev found that certain combinatorial-topological cyclicities of classical random variables form a necessary and sufficient condition for the constraints that are now known as Bell-CHSH inequalities. These constraints, however, must not be linked to the observables of quantum theory nor to the actual EPRB experiments for a variety of reasons, including the existence of continuum-related variables and appropriate considerations of symmetry. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
Show Figures

Figure 1

Back to TopTop